These conclusions are again
summarized in Table REF .
The role of an observable which would be manifestly self-adjoint in
\({\cal H}_{phys}\)  can now be played by \(Z_1\) , by the product \(Z_2\,Z_1\)  and
by the product \(Z_3\,Z_2\,Z_1\) , but not by the product \(Z_3Z_2\)  or
by the operators \(Z_2\)  or \(Z_3\)  standing alone (the proofs are
analogous to the ones given above). Thus, in the
more conventional notation
one can conclude that the “hidden Hermiticity” (i.e., the
observability) status holds for \(H\)  and for the charge \({\cal C}=Z_1\) . The observability status of the “other charge” \({\cal D}=Z_2\) 
(i.e., the validity of condition \({\cal D}^\dagger \,\Theta _3=\Theta _3\,{\cal D}\) ) could only be achieved,
in a sufficiently elementary manner,
under an additional commutativity requirement \({\cal D}\,{\cal C}={\cal C}\,{\cal D}\) . Otherwise,
the observability status of \({\cal D}\)  would require
a complicated analysis as sampled
in Ref. [1]}.
